## Mathematics
16:642:611
Selected
Topics in Applied Mathematics: Variational
Inequalities, Obstacle and Free Boundary Problems in Mathematical
Finance

### Instructor

Paul
Feehan
### Questions
about
the
course?

Please feel welcome to contact
me!
### Schedule

The course will be offered during
the Fall 2011 semester (Tuesdays and Fridays 10:20-11:40 in Hill 425, Wednesdays 10:20-11:40
in SEC 204). The
Friday meeting times are reserved for make-up classes, optional
problem sessions, and guest lectures. The Tuesday meeting time may be
moved
slightly earlier (10:00-11:20)
to avoid conflict with a seminar later that morning.
### Course
summary

The goal of the course is to
introduce graduate students to the theory required to do research on
variational inequalities, obstacle problems, and free boundary problems
and their applications to pure and applied mathematics. To accomplish
this goal,

we plan to cover the
following topics:
- Existence, uniqueness, and global regularity results for
solutions to elliptic obstacle problems (stationary variational
inequalities).
- Optimal regularity of solutions to elliptic obstacle problems
near the free boundary.
- Regularity of the free boundary for solutions to elliptic
obstacle problems.

- Existence, uniqueness, and global regularity results for
solutions to parabolic obstacle problems (evolutionary variational
inequalities).
- Optimal regularity
of solutions to parabolic obstacle problems near the free boundary.
- Regularity of the
free boundary for solutions to parabolic obstacle problems.
- Weighted Sobolev
spaces and applications to degenerate elliptic and parabolic partial differential equations and
obstacle problems.
- Applications: sample problems will be drawn from
mathematical finance, through additional problems for other
applications will be considered depending on the
interests of course participants.

### Optional third weekly meeting

The optional

third weekly meeting
times are reserved for make-up classes, problem sessions, and guest
lectures. Examples of guest lectures may include

- Presentations by other faculty members on topics related to the
course. Examples include numerical solution of variational
inequalities, degenerate partial differential equations and weighted
Sobolev spaces, specific obstacle problems in applied mathematics,
relationship with probability theory and stochastic processes, more
applications to mathematical finance, among others.
- Presentations by students on topics related to the course.
Examples include thesis research problems or solutions to homework
problems (all homework assignments are optional).

### Audience

Second and higher-year doctoral students;
second-year master's students with required mathematics background;
junior faculty members.

### Pre-requisites

A one-semester graduate level course
on real analysis (for example, Math 16:640:501) covering measure
theory, Hilbert spaces, and Banach spaces. An undergraduate course on
real analysis (for example, Math 01:640:411-412) may suffice for
motivated graduate students.
The following short text

is
recommended reading
prior to the course for anyone who has not taken a
graduate level class on partial differential equations:

- Q. Han and F. Lin, Elliptic
partial differential equations, Courant Lecture Notes, American
Mathematical Society, Providence, RI, 2011.

### Co-requisites

A one-semester graduate level course
on partial differential equations (for example, Math 16:640:517 or a
similar course based on the text by Evans) is recommended, but not
required. The course will be self-contained in order to accommodate
beginning second-year students who wish to explore potential research
topics in partial differential equations and their applications.
### Non-requisites

- Do I need to know anything
about probability theory or stochastic processes? No! I will
provide references and additional problems to students interested in
applications to probability theory and Markov processes, but
probability theory and stochastic processes will not be covered in the
course. Students interested in probability theory and stochastic
processes should consider also taking Math 16:642:591 Topics in
Probability and Ergodic Theory I.
- Do I need to know anything
about mathematical finance? No! I will provide references and
additional problems to students interested in applications to
mathematical finance (and other applications), but mathematical
finance itself will not be covered in
the course.
- Should I have taken an
undergraduate level course on partial differential equations?
While prior exposure to partial differential equations through a course
such as Math 01:640:423 has some marginal benefit, that is not required.

### Textbooks

- A. Petrosyan, H. Shahgholian, N. Ural'tseva, Regularity of free boundaries in obstacle
type problems, publication expected in 2011.
- J-F. Rodrigues, Obstacle problems in mathematical physics
, North-Holland, New York, 1987.

### Primary
reference
texts

- A. Bensoussan and J. L. Lions, Applications
of
variational
inequalities in stochastic control,
North-Holland, New York, 1982.
- L. Caffarelli and S. Salsa, A
geometric approach to free boundary problems, Graduate Studies
in Mathematics, vol. 68, American Mathematical Society, Providence, RI,
2005.
- A. Friedman, Variational
principles and free boundary problems, Dover, New York, 2010.
- D. Kinderlehrer and G. Stampacchia, An introduction to variational
inequalities and their applications, Academic, New York, 1980.
- V. I. Maz'ya, Sobolev spaces: with applications to elliptic
partial differential equations, Springer, New York, 2011.
- G. M. Troianiello, Elliptic differential equations and
obstacle problems, Plenum, New York, 1987.

###
Supplementary reference texts

- L. C. Evans, Partial
differential equations, second edition, American Mathematical
Society, Providence, RI, 2010.
- A. Friedman, Partial
differential equations, Dover, New York, 2008.
- A. Friedman, Partial
differential equations of parabolic type, Dover, New York, 2008.
- D. Gilbarg and N. S. Trudinger, Elliptic
partial
differential
equations of second order, Springer, New
York, 2001.

- Q. Han and F. Lin, Elliptic
partial differential equations, Courant Lecture Notes, American
Mathematical Society, Providence, RI, 2011.
- N. V. Krylov, Lectures on
elliptic and parabolic equations in Hölder spaces, Graduate
Studies in Mathematics, vol. 12, American Mathematical Society,
Providence, RI, 1996.
- N. V. Krylov, Lectures on
elliptic and parabolic equations in Sobolev spaces, Graduate
Studies in Mathematics, vol. 96, American Mathematical Society,
Providence, RI, 2008.

- G. M. Lieberman, Second order
parabolic differential equations, World Scientific, River Edge,
NJ, 1996.
- R. E. Showalter, Monotone
operators in Banach space and nonlinear partial differential equations,
American
Mathematical Society, Providence, RI, 1996. (Free e-copy from
AMS).

### Reference articles

Research articles relevant to the course will be provided, including
those by the following authors:

- L. Caffarelli and collaborators.
- P. Daskalopoulos and collaborators.
- P. Feehan and collaborators.
- P. Laurence and collaborators.

- H. Shahgholian and collaborators.

### Grading

The course is intended for
second and higher-year graduate students primarily interested in
research in partial
differential equations and applications, so there will be no exams or
required homework.
Optional weekly written homework and reading assignments will be
provided and homework assignments may be discussed during the optional
third weekly meeting periods.
### Syllabus,
lecture
notes,
homework problems, and reading assignments

The weekly syllabus, readings,
homework problems, lecture notes and research articles will be provided
on Sakai to registered
participants.